Phasor Relationships for Circuit Elements 7 4 Prof

Phasor Relationships for Circuit Elements (7. 4) Prof. Phillips April 16, 2003 lecture 20 1

Phasor Relationships for Circuit Elements • Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor. • A complex exponential is the mathematical tool needed to obtain this relationship. lecture 20 2

I-V Relationship for a Resistor + i(t) v(t) R – Suppose that i(t) is a sinusoid: i(t) = IM ej(wt+q) Find v(t) lecture 20 3

Computing the Voltage lecture 20 4

Class Example lecture 20 5

I-V Relationship for a Capacitor + i(t) v(t) C – Suppose that v(t) is a sinusoid: v(t) = VM ej(wt+q) Find i(t) lecture 20 6

Computing the Current lecture 20 7

Phasor Relationship • Represent v(t) and i(t) as phasors: V = VM q I = jw. C V • The derivative in the relationship between v(t) and i(t) becomes a multiplication by jw in the relationship between V and I. lecture 20 8

Example v(t) = 120 V cos(377 t + 30 ) C = 2 m. F • What is V? • What is I? • What is i(t)? lecture 20 9

Class Example lecture 20 10

I-V Relationship for an Inductor + i(t) v(t) L – V = jw. L I lecture 20 11

Example i(t) = 1 m. A cos(2 p 9. 15 • 107 t + 30 ) L = 1 m. H • What is I? • What is V? • What is v(t)? lecture 20 12

Class Example lecture 20 13

Circuit Element Phasor Relations (ELI and ICE man) lecture 20 14

Phasor Diagrams • A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). • A phasor diagram helps to visualize the relationships between currents and voltages. lecture 20 15

An Example 2 m. A 40 + + 1 m. F w = 377 1 k. W – VC + – V VR – lecture 20 16

An Example (cont. ) I = 2 m. A 40 VR = 2 V 40 VC = 5. 31 V -50 V = 5. 67 V -29. 37 lecture 20 17

Phasor Diagram Imaginary Axis Real Axis V VC VR lecture 20 18

MATLAB Exercise • Let’s use MATLAB to plot an ac current and voltage, and then to graphically determine the lead-lag relationship • Start MATLAB on your computer • We begin by creating a time vector >> t = 0 : 0. 0005 : 0. 025; • Next, we create the voltage and current >> vt = 170 * cos(377*t+10*pi/180); >> it = 100 * cos(377*t-65*pi/180); lecture 20 19

MATLAB Exercise • Now we will graph v(t) and i(t) >> plot(t, vt, 'b', t, it, 'r--'); >> xlabel('Time (sec)'); >> ylabel('Voltage (Volts) or Current (Amps)'); >> title('Household AC Voltage-Current'); >> legend('v(t)=170 cos(377 t+10)', 'i(t)=100 cos(377 t-65)'); lecture 20 20

MATLAB Exercise • From the graphs created: – Determine whether the current leads the voltage, or vice versa – Determine the amount of lead by the current or voltage • Compare the voltage-current lead-lag relationship obtained by graphical means above to an analytic solution which you should be able to compute lecture 20 21